Ramsey properties of random hypergraphs
Let \(K^{(k)} (n,p)\) denote the random \(k\)-uniform hypergraph obtained by the independent inclusion of each of the \({n\choose k}\) \(k\)-subsets with probability \(p\). If \(F\) and \(G\) are two \(k\)-uniform hypergraphs, the Ramsey theory arrow notation, \(F\to(G)^e_r\), stands for the statement: For every partition of the edges of \(F\) into \(r\) classes, at least one of the classes contains a copy of \(G\). The main result of this paper is Theorem 1.4: There exists an absolute constant \(C>0\) such that \(\lim_{n\to\infty} P(K^{(3)} (n,p)\to (K_4^{(3)})_2^{e}) =1\) for \(p>Cn^{-{1 \over 3}}\).
scientific article; zbMATH DE number 1123546